reserve u,v,x,y,z,X,Y for set;
reserve r,s for Real;
reserve N for non empty ConjNormAlgStr;
reserve a,a1,a2,b,b1,b2 for Element of N;

theorem Th9:
  N is add-associative right_zeroed right_complementable well-conjugated
  reflexive scalar-distributive scalar-unital vector-distributive
  left-distributive
  implies for a holds a*' * a = ||.a.||^2 * 1.N
  proof
    assume that
A1: N is add-associative right_zeroed right_complementable and
A2: N is well-conjugated and
A3: N is reflexive and
A4: N is scalar-distributive scalar-unital vector-distributive
    left-distributive;
    let a;
    per cases;
    suppose a is non zero;
      hence a*' * a = ||.a.||^2 * 1.N by A2,Def3;
    end;
    suppose
A5:   a is zero;
      then
A6:   a = 0.N & a*' = 0.N by A2;
A7:   ||.a.||^2 = 0 by A3,A5;
      0 * 1.N = 0.N by A1,A4,RLVECT_1:10;
      hence thesis by A1,A6,A4,A7;
    end;
  end;
